What I Learned While On Blog Hiatus (3)

When last we left off, we had seen the putative basis for a linearly evolving non-dimensional shear d(x) which could be mapped onto the x-y plane overlaid on a solar active region. See e.g. the diagram in:

http://brane-space.blogspot.com/2012/05/what-i-learned-while-on-blog-hiatus-2.html

As we change the (dimensionless) shear through progressive numerical substitutions, the shear gradient m increases so that the dimensionless shear kφ  increases to a maximum of 1, then decreases. Likewise, in doing this, one finds the line y = mx altering with m to yield evolving arcade footpoint axes (in relation to the potential or y = 0 configuration) of the arcade when in the potential or current free condition.


 We are left with incorporating three functions:

1) d(x) = x sinh b

2) B n (x) = x / R o [ B _f  ( R o ) ]


and

3) v(φ,W) = a sin (φ ) exp [1 - 2[φ ]/ W



into one partial differential equation, e.g.

u_t + au_x = 0

which is amenable to an efficiently generated numerical simulation algorithm.
Note first that (2) can be recast as:

x =  B _N (x) [R _o  [ B_f   ( R _o ) ]


And  we can choose as a first value for a: a =  dx(kφ,W) /dt

where v (φ, W) in (2) is now dx(kφ,W) /dt


The final functional solution which we obtain for possible discretization after obtaining its PDE and after substituting for x in x sinh b, is:

u(x,t) =  [ B n (x) [R _o [ B_f  ( R _o ) ]  sinh b +  dx(kφ,W)/dt

And we note that the dimensionless shear kφ is actually a time -dependent function of the gradient, e.g. m(t).

Alas, the complexity of the (differentiated) equation became evident as I went through several initial runs merely testing for errors and stability for two methods, the finite difference and spectral method (which uses Fourier series)).  What did I learn ?   Well, that I bit off a bit more than I could chew in a 2 1/2 week project. But hey, at least I made a start!